Pancyclic out-arcs of a vertex in tournaments
Proceedings of the 5th Twente workshop on on Graphs and combinatorial optimization
An s-strong tournament with s≥3 has s + 1 vertices whose out-arcs are 4-pancyclic
Discrete Applied Mathematics
Note: The number of vertices whose out-arcs are pancyclic in a 2-strong tournament
Discrete Applied Mathematics
The number of pancyclic arcs in a k-strong tournament
Journal of Graph Theory
Digraphs: Theory, Algorithms and Applications
Digraphs: Theory, Algorithms and Applications
Each 3-strong Tournament Contains 3 Vertices Whose Out-arcs Are Pancyclic
Graphs and Combinatorics
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Yao et al. (Discrete Appl Math 99 (2000), 245–249) proved that every strong tournament contains a vertex u such that every out-arc of u is pancyclic and conjectured that every k-strong tournament contains k such vertices. At present, it is known that this conjecture is true for k = 1, 2, 3 and not true for k⩾4. In this article, we obtain a sufficient and necessary condition for a 4-strong tournament to contain exactly three out-arc pancyclic vertices, which shows that a 4-strong tournament contains at least four out-arc pancyclic vertices except for a given class of tournaments. Furthermore, our proof yields a polynomial algorithm to decide if a 4-strong tournament has exactly three out-arc pancyclic vertices. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: Natural Science Foundation of China; Contract grant number: 11026162 (to Q. G., S. L., and R. L.); Contract grant sponsor: Natural Science Foundation for Young Scientists of Shanxi Province, China; Contract grant number: 2011021004 (to Q. G., S. L., and R. L.).)